VALUES IN A CANONICAL ENSEMBLE. 47 



the multiple integral may be resolved into the product of 

 integrals 



C. . . C 



m v dz v . (121) 



This shows that the probability that the configuration lies 

 within any given limits is independent of the velocities, 

 and that the probability that any component velocity lies 

 within any given limits is independent of the other component 

 velocities and of the configuration. 

 Since 



* 2 



f 4 V>, <& = vz^, ( 122 > 



I/ 00 



and 



J 



e 2 m* dx! = V^Ti-mx 8 , ( 123 > 



the average value of the part of the kinetic energy due to the 

 velocity x 19 which is expressed by the quotient of these inte- 

 grals, is J <H). This is true whether the average is taken for 

 the whole ensemble or for any particular configuration, 

 whether it is taken without reference to the other component 

 velocities, or only those systems are considered in which the 

 other component velocities have particular values or lie 

 within specified limits. 



The number of coordinates is 3 v or n. We have, therefore, 

 for the average value of the kinetic energy of a system 



e p = ! = w. (124) 



This is equally true whether we take the average for the whole 

 ensemble, or limit the average to a single configuration. 



The distribution of the systems with respect to their com- 

 ponent velocities follows the * law of errors ' ; the probability 

 that the value of any component velocity lies within any given 

 limits being represented by the value of the corresponding 

 integral in (121) for those limits, divided by (2 TT m )*, 



